# 8.2 Decimals: addition and subtraction of decimals

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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses how to add and subtract decimals. By the end of the module students should understand the method used for adding and subtracting decimals, be able to add and subtract decimals and be able to use the calculator to add and subtract decimals.

## Section overview

• The Logic Behind the Method
• The Method of Adding and Subtracting Decimals
• Calculators

## The logic behind the method

Consider the sum of 4.37 and 3.22. Changing each decimal to a fraction, we have

$4\frac{\text{37}}{\text{100}}+3\frac{\text{22}}{\text{100}}$ Performing the addition, we get

$\begin{array}{ccc}4\text{.}\text{37}+3\text{.}\text{22}=4\frac{\text{37}}{\text{100}}+3\frac{\text{22}}{\text{100}}& =& \frac{4\cdot \text{100}+\text{37}}{\text{100}}+\frac{3\cdot \text{100}+\text{22}}{\text{100}}\hfill \\ & =& \frac{\text{437}}{\text{100}}+\frac{\text{322}}{\text{100}}\hfill \\ & =& \frac{\text{437}+\text{322}}{\text{100}}\hfill \\ & =& \frac{\text{759}}{\text{100}}\hfill \\ & =& 7\frac{\text{59}}{\text{100}}\hfill \\ & =& \text{seven and fifty-nine hundredths}\\ & =& 7.59\hfill \end{array}$

Thus, $4\text{.}\text{37}+\text{3}\text{.}\text{22}=\text{7}\text{.}\text{59}$ .

## The method of adding and subtracting decimals

When writing the previous addition, we could have written the numbers in col­umns.

$\begin{array}{c}\hfill 4.37\\ \hfill \underline{+3.22}\\ \hfill 7.59\end{array}$

This agrees with our previous result. From this observation, we can suggest a method for adding and subtracting decimal numbers.

## Method of adding and subtracting decimals

1. Align the numbers vertically so that the decimal points line up under each other and the corresponding decimal positions are in the same column.
2. Add or subtract the numbers as if they were whole numbers.
3. Place a decimal point in the resulting sum or difference directly under the other decimal points.

## Sample set a

Find the following sums and differences.

$9\text{.}\text{813}+2\text{.}\text{140}$

\begin{array}{cc}\hfill 9.813& \text{The decimal points are aligned in the same column.}\hfill \\ \hfill \underline{+2.140}& \\ 11.953& \end{array}

$\text{841}\text{.}\text{0056}+\text{47}\text{.}\text{016}+\text{19}\text{.}\text{058}$

To insure that the columns align properly, we can write a 0 in the position at the end of the numbers 47.016 and 19.058 without changing their values.

$1.314-0.58$

$16.01-7.053$

Find the sum of 6.88106 and 3.5219 and round it to three decimal places.

We need to round the sum to the thousandths position. Since the digit in the position immediately to the right is 9, and 9>5, we get

10.403

Wendy has $643.12 in her checking account. She writes a check for$16.92. How much is her new account balance?

To find the new account balance, we need to find the difference between 643.12 and 16.92. We will subtract 16.92 from 643.12.

After writing a check for $16.92, Wendy now has a balance of$626.20 in her checking account.

## Practice set a

Find the following sums and differences.

$3\text{.}\text{187}+\text{2}\text{.}\text{992}$

6.179

$\text{14}\text{.}\text{987}-\text{5}\text{.}\text{341}$

9.646

$0\text{.}\text{5261}+\text{1}\text{.}\text{0783}$

1.6044

$1.06-1.0535$

0.0065

$\text{16,521}\text{.}\text{07}+\text{9,256}\text{.}\text{15}$

25,777.22

Find the sum of 11.6128 and 14.07353, and round it to two decimal places.

25.69

## Calculators

The calculator can be useful for finding sums and differences of decimal numbers. However, calculators with an eight-digit display cannot be used when working with decimal numbers that contain more than eight digits, or when the sum results in more than eight digits. In practice, an eight-place decimal will seldom be encoun­tered. There are some inexpensive calculators that can handle 13 decimal places.

## Sample set b

Use a calculator to find each sum or difference.

42.0638 + 126.551

 Display Reads Type 42.0638 42.0638 Press + 42.0638 Type 126.551 126.551 Press = 168.6148

The sum is 168.6148.

Find the difference between 305.0627 and 14.29667.

 Display Reads Type 305.0627 305.0627 Press — 305.0627 Type 14.29667 14.29667 Press = 290.76603

The difference is 290.76603

51.07 + 3,891.001786

Since 3,891.001786 contains more than eight digits, we will be unable to use an eight-digit display calculator to perform this addition. We can, however, find the sum by hand.

$\begin{array}{c}\hfill 51.070000\\ \hfill \underline{3891.001786}\\ 3942.071786\end{array}$

The sum is 3,942.071786.

## Practice set b

Use a calculator to perform each operation.

$4\text{.}\text{286}+\text{8}\text{.}\text{97}$

13.256

$452.0092-392.558$

59.4512

Find the sum of 0.095 and 0.001862

0.096862

Find the difference between 0.5 and 0.025

0.475

Find the sum of 2,776.00019 and 2,009.00012.

Since each number contains more than eight digits, using some calculators may not be helpful. Adding these by “hand technology,” we get 4,785.00031

## Exercises

For the following 15 problems, perform each addition or subtraction. Use a calculator to check each result.

$1\text{.}\text{84}+7\text{.}\text{11}$

8.95

$\text{15}\text{.}\text{015}-6\text{.}\text{527}$

$\text{11}\text{.}\text{842}+\text{28}\text{.}\text{004}$

39.846

$3\text{.}\text{16}-2\text{.}\text{52}$

$3\text{.}\text{55267}+8\text{.}\text{19664}$

11.74931

$0\text{.}\text{9162}-0\text{.}\text{0872}$

$\text{65}\text{.}\text{512}-8\text{.}\text{3005}$

57.2115

$\text{761}\text{.}\text{0808}-\text{53}\text{.}\text{198}$

$4\text{.}\text{305}+2\text{.}\text{119}-3\text{.}\text{817}$

2.607

$\text{19}\text{.}\text{1161}+\text{27}\text{.}\text{8014}+\text{39}\text{.}\text{3161}$

$0\text{.}\text{41276}-0\text{.}\text{0018}-0\text{.}\text{00011}$

0.41085

$2\text{.}\text{181}+6\text{.}\text{05}+1\text{.}\text{167}+8\text{.}\text{101}$

1.0031+6.013106+0.00018+0.0092+2.11

9.135586

$\text{27}+\text{42}+9\text{.}\text{16}-0\text{.}\text{1761}+\text{81}\text{.}6$

$\text{10}\text{.}\text{28}+\text{11}\text{.}\text{111}+0\text{.}\text{86}+5\text{.}1$

27.351

For the following 10 problems, solve as directed. A calculator may be useful.

Add 6.1121 and 4.916 and round to 2 decimal places.

Add 21.66418 and 18.00184 and round to 4 decimal places.

39.6660

Subtract 5.2121 from 9.6341 and round to 1 decimal place.

Subtract 0.918 from 12.006 and round to 2 decimal places.

11.09

Subtract 7.01884 from the sum of 13.11848 and 2.108 and round to 4 decimal places.

A checking account has a balance of $42.51. A check is written for$19.28. What is the new balance?

$23.23 A checking account has a balance of$82.97. One check is written for $6.49 and another for$39.95. What is the new balance?

A person buys $4.29 worth of hamburger and pays for it with a$10 bill. How much change does this person get?

$5.71 A man buys$6.43 worth of stationary and pays for it with a $20 bill. After receiving his change, he realizes he forgot to buy a pen. If the total price of the pen is$2.12, and he buys it, how much of the \$20 bill is left?

A woman starts recording a movie on her video cassette recorder with the tape counter set at 21.93. The movie runs 847.44 tape counter units. What is the final tape counter reading?

869.37

## Exercises for review

( [link] ) Find the difference between 11,206 and 10,884.

( [link] ) Find the product, $\text{820}\cdot \text{10,000}$ .

8,200,000

( [link] ) Find the value of $\sqrt{\text{121}}-\sqrt{\text{25}}+{8}^{2}+\text{16}÷{2}^{2}$ .

( [link] ) Find the value of $8\frac{1}{3}\cdot \frac{\text{36}}{\text{75}}÷2\frac{2}{5}$ .

( [link] ) Round 1.08196 to the nearest hundredth.

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